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computational_science/graduate/compressed_sensing_and_sparse_optimization/advanced_topics/sparsity_in_machine_learning/convolutional_sparse_coding/Feynman/zh-CN/卷积稀疏编码/MainContent.md

@@ -7,6 +7,9 @@
 
 准备好揭开信号背后隐藏的结构之美了吗？让我们首先深入其内部，探寻CSC运行的基本原理与精巧机制。
 
+The following chapters will guide you from the foundational mathematics to real-world problem-solving. In "Principles and Mechanisms," we will dissect the mathematical core of the CSC model, exploring its formulation, the crucial concept of [shift-invariance](@entry_id:754776), its connection to compressed sensing theory, and the non-convex challenges of [dictionary learning](@entry_id:748389). Next, "Applications and Interdisciplinary Connections" will showcase the versatility of CSC, demonstrating how the model is extended and applied to solve complex problems in image processing, [computational imaging](@entry_id:170703), and [geophysics](@entry_id:147342). Finally, "Hands-On Practices" will offer you the chance to solidify your understanding by tackling concrete implementation challenges, from calculating algorithmic parameters to analyzing model trade-offs.
+
+
 ## 原理与机制
 
 在上一章中，我们对卷积[稀疏编码](@entry_id:180626)（Convolutional Sparse Coding, CSC）有了初步的印象。现在，让我们像物理学家一样，深入其内部，探寻其运行的基本原理与精巧机制。我们将一起踏上一段旅程，去发现这个模型背后蕴含的深刻洞见、数学之美以及它与我们看待世界方式的奇妙联系。

keyword/g/g_invariant_subspace/Feynman/zh-CN/G-不变子空间/MainContent.md

@@ -9,6 +9,14 @@
 
 本章的目标是深入探究这些表示的内部工作机制。我们希望理解一个由对称性支配的复杂系统如何能够被分解为其最简单、最基本的部分。正如化学家将分子分解为原子一样，我们也希望将一个[表示分解](@article_id:299509)为其“原子”组分。整个过程的关键在于**$G$-不变子空间**的概念。
 
+### The Decomposition Theorem: Maschke's Great Insight
+
+It turns out that for a huge and important class of groups and representations, the answer is a resounding *yes*. This wonderful guarantee comes from **Maschke's Theorem**. In its common form, it states:
+
+> Let $G$ be a **finite group** and let $V$ be a finite-dimensional representation of $G$ over a field $F$ whose characteristic does not divide the order of $G$. If $W$ is a G-[invariant subspace](@article_id:136530) of $V$, then there exists another G-invariant subspace $U$ such that $V = W \oplus U$.
+
+This means that every invariant subspace has an invariant complement. This property is called **[complete reducibility](@article_id:143935)**. If we have it, we can take any [reducible representation](@article_id:143143), split it into an [invariant subspace](@article_id:136530) and its invariant complement, and then repeat the process on those smaller pieces until all we're left with are irreducible "atoms".
+
 ### 对称性的DNA：不变子空间
 
 假设我们有一个[向量空间](@article_id:297288) $V$，你可以将其视为我们系统所有可能状态的“宇宙”。群 $G$ 的一个表示是作用于 $V$ 中向量的一系列[线性变换](@article_id:376365) $\rho(g)$，每个群元素 $g \in G$ 对应一个。一个**不变子空间**是这个宇宙中一个非常特殊的地方。它是 $V$ 内的一个子空间 $W$，在群作用下是“自洽的”。无论你从 $W$ 中选取哪个向量 $w$，也无论你对其施加哪个对称操作 $\rho(g)$，得到的向量 $\rho(g)w$ 都*永远不会*离开 $W$。该子空间在整个群的作用下是封闭的。